Problem Solving

BTGmoderatorDC wrote: ↑

Wed Dec 02, 2020 5:58 pm

Consider the following three triangles

I. a triangle with sides 6-9-10
II. a triangle with sides 8-14-17
III. a triangle with sides 5-12-14

Which of the following gives a complete set of the triangles that have at least one obtuse angle, that is, an angle greater than 90°?

(A) I
(B) II
(C) III
(D) I & II
(E) II & III


OA E

Solution:

Recall that if a, b, and c are the side lengths of a triangle where c is the longest side length and:

1) a^2 + b^2 = c^2, then the triangle is a right triangle.

2) a^2 + b^2 > c^2, then the triangle is an acute triangle.

3) a^2 + b^2 < c^2, then the triangle is an obtuse triangle.

Since only obtuse triangles could have an obtuse angle, we are determining whether the given triangles are obtuse or not. Of course, if they are, case 3 will be satisfied. Otherwise, it will not.

I. a triangle with sides 6-9-10

We see that 10 is the longest side and we need to determine whether 6^2 + 9^2 < 10^2.

36 + 81 < 100 ?

117 < 100 ?

We see that 117 is NOT less than 100, so the triangle with sides 6-9-10 can’t be an obtuse triangle and thus it won’t contain an obtuse angle.

II. a triangle with sides 8-14-17

We see that 17 is the longest side and we need to determine whether 8^2 + 14^2 < 17^2.

64 + 196 < 289 ?

260 < 289 ?

We see that 260 is INDEED less than 289, so the triangle with sides 8-14-17 is an obtuse triangle and thus it will contain an obtuse angle.

III. a triangle with sides 5-12-14

We see that 14 is the longest side and we need to determine whether 5^2 + 12^2 < 14^2.

25 + 144 < 196 ?

169 < 196 ?

We see that 169 is INDEED less than 196, so the triangle with sides 5-12-14 is an obtuse triangle and thus it will contain an obtuse angle.

Answer: E